I think it's fair to say that a lot (perhaps most) of basic research in theoretical physics these days takes place within the pure mathematics departments of universities. I suspect the reason for this is that the amount of maths necessary to tackle these subjects could fill an entire career. If I'm going to understand this field at any substantial level it looks like I'm going to have to take a heck of a lot of maths. To put things into context, I'm third undergrad at an Australian university majoring in physics/pure maths. I do a bit of research into the the theory of complex colloidal plasmas with a couple of publications to my name. At the moment I'm planning on doing Honours (fourth year) followed by a PhD in the same field. I'd like to be able to understand current research in the more theoretical areas such as strings and loop quantum gravity. My current mathematical knowledge is OK but not excellent. I've done PDEs, vector calculus and group theory at second year level. This year (third year), I've so far done metric spaces (topology) and field theory. I will be doing differential geometry and Lagrangian and Hamiltonian dynamics next semester. I've done no pure analysis courses (unless you count metrics). The following pure mathematics courses (in addition to advanced third year courses) are offered in fourth year Functional analysis Partial differential equations Algebraic topology Algebraic geometry Commutative algebra Representations of the symmetric group http://www.maths.usyd.edu.au/u/UG/HM/pure2007.pdf I understand that functional analysis is basically quantum mechanics and that alg topology has some applications to LQG. I also understand how useful group theory can be in physics as I've encountered it in my research on plasma physics. Would anyone be able to rate these in terms of their usefulness for theoretical physics? Please keep in mind that by majoring in physics I'm limiting the amount of maths I can do to probably 2 of these. Thanks in advance James.